I am new to nodeJS,
I am trying to install mysql package of nodeJS as
>node
>npm install mysql
and also with
>npm install node-mysql
but getting error as
npm should be run out side of the node repl in your normal shell
so i run the same command in side nodeJS directory in program file
still getting error check images

This error is indicating that the path referred to in the screenshot above (.../AppData/Roaming/npm) is not available (doesn't exist or isn't accessible by your user).
Can you make sure there is a folder npm at this path:
c:\Users\mayank.bhadauria\appdata\roaming
If not, please create it and then try npm install node-mysql again.